Uniform Circular Motion Practice Problems

A toy is made using a frictionless spherical shell with an inner radius of 35 cm and a tiny pearl of mass 8.0 g. The pearl lies on the inner shell, as shown in the figure. The shell can rotate about a fixed vertical axis at various angular speeds. When playing, a child remarks that the pearl moves from the bottom of the shell to a higher position only when the shell rotates at an angular speed greater than ω_a. The line that connects the center of the shell to the pearl forms an angle α with the vertical. Find ω_a in revolutions per minute.

A 75-cm-long, massless, horizontal bar is connected at one end to an electric motor that rotates the bar at a constant angular speed of 48 rpm. A 250-gram ball is attached to the other end. Calculate the magnitude of the ball's velocity.

A weather satellite is in geostationary orbit, 36000 km above the equator. The satellite rotates around the earth at the same rate as the earth does. The satellite monitors the weather evolution of the same region of the earth using ultrasensitive detectors. Calculate the speed and acceleration of the satellite. The Earth's equatorial radius is 6378 km.

An extrasolar planet orbits around its star in a uniform circular motion of radius 1.43 × 10³ km. The planet completes one orbit in 21.3 days. What is the magnitude of the planet's angular velocity?

A meteorological satellite moves around our planet in a circular, uniform motion with a radius of 7242 km. Its orbital period is 101 min. Calculate the speed of the satellite.

A ceiling fan has a radius of 60 cm and 1200 rpm at full speed. A particle of mass 1 g sticks at the outer edge of the fan blade. Calculate the radial acceleration experienced by the particle when the fan runs at full speed.

A wind turbine contains 120 ft long blades and a propeller that rotates at 10 rpm. Calculate the linear speed at the blade tips, in ft/min.

The radius of Mars is 3,389.5 km and rotates once every 24.6 hours. The radial acceleration at the equator must be larger than g for items to leave the surface of the planet and enter space. What would be the needed rotational period on Mars for this to happen? (g on the surface of mars is 3.7 m/s²)

The radius of the earth is 6371 km, and a 200-kilogram satellite is launched into an orbit 700 km above the planet's surface. The satellite takes 1.64 hours to complete one orbit. Assuming a circular orbit, what is the radial acceleration of the satellite?

Centrifuges conveniently create any desired value of acceleration using rotation. Scientists use centrifuges to study the effects of high acceleration on human beings. However, large accelerations are not so nice to human beings. A 1.65 m tall person is strapped onto a 7.85 long centrifuge arm with their head at the outermost end. Calculate the difference in acceleration experienced by the head and feet of the person if they are subjected to a maximum acceleration of 9g.

Scientists use centrifuges to study the effects of large accelerations on human beings. The centrifuges are a convenient tool because of their ability to produce large accelerations. In a typical experiment, a person with a height of 1.58 m is centrifuged using a centrifuge arm that is 6.83 m long, with the person's head located at the outermost end of the centrifuge arm. Calculate the speed of the person's head if they are subjected to a maximum acceleration of 9g.

A plane flies in a completely horizontal circle at a constant speed of 300 km/h. Calculate the diameter of the circle if the plane's radial acceleration is equal to twice the gravitational acceleration.